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pro vyhledávání: '"Christian Axler"'
Autor:
Christian Axler
Publikováno v:
The Ramanujan Journal.
Let $$\sigma (n)$$ σ ( n ) denote the sum of divisors function of a positive integer n. Robin proved that the Riemann hypothesis is true if and only if the inequality $$\sigma (n) < \textrm{e}^{\gamma }n \log \log n$$ σ ( n ) < e γ n log log n hol
Autor:
Jean-Louis Nicolas, Christian Axler
Publikováno v:
Acta Arithmetica.
Autor:
Christian Axler
Publikováno v:
Journal de Théorie des Nombres de Bordeaux. 31:293-311
In this paper we establish a general asymptotic formula for the sum of the first n prime numbers, which leads to a generalization of the most accurate asymptotic formula given by Massias and Robin in 1996.
Comment: 5 pages
Comment: 5 pages
Autor:
Christian Axler
Publikováno v:
Funct. Approx. Comment. Math. 63, no. 1 (2020), 67-93
The $n$th Ramanujan prime is the smallest positive integer $R_n$ such that for all $x \geq R_n$ the interval $(x/2, x]$ contains at least $n$ primes. In this paper we undertake a study of the sequence $(\pi(R_n))_{n \geq 1}$, which tells us where the
Autor:
Christian Axler
Publikováno v:
Bulletin of the Australian Mathematical Society. 96:374-379
Robin’s criterion states that the Riemann hypothesis is true if and only if $\unicode[STIX]{x1D70E}(n) for every positive integer $n\geq 5041$. In this paper we establish a new unconditional upper bound for the sum of divisors function, which impro
Autor:
Christian Axler
Publikováno v:
Mediterranean Journal of Mathematics. 15
In this paper, we establish explicit upper and lower bounds for the ratio of the arithmetic and geometric means of the first n prime numbers, which improve the current best estimates. Furthermore, we prove several conjectures related to this ratio st
Autor:
Christian Axler, Thomas Leßmann
Publikováno v:
The American Mathematical Monthly. 124:642
In this paper we compute the explicit values for the first k-Ramanujan prime for every k ≥ 1.000040690557321 by using an elegant characterization of the first k-Ramanujan prime, which is established in this paper, and a recent result concerning the
Autor:
Christian Axler
In this paper we establish several results concerning the generalized Ramanujan primes. For $n\in\mathbb{N}$ and $k \in \mathbb{R}_{> 1}$ we give estimates for the $n$th $k$-Ramanujan prime which lead both to generalizations and to improvements of th
Externí odkaz:
https://explore.openaire.eu/search/publication?articleId=doi_dedup___::cea1fcf1384ec6df4a1a06f61aaefd56
http://arxiv.org/abs/1401.7179
http://arxiv.org/abs/1401.7179